We know that each three-dimensional shape has faces, edges, and vertices.

**Faces:** A flat surface.**Edges:** A point where two faces meet each other.**Vertices:** A corner where edges meet.

A three-dimensional shape

which has sharp vertices, flat polygonal faces,

and straight edges is called a polyhedron.

**1) What Are the Types of Polyhedrons?**

There are two types of polyhedrons, namely:

- Convex polyhedron
- Concave polyhedron

**2) What is Convex Polyhedron?**

A polyhedron whose faces, edges and vertices do not intersect each other is called a convex polyhedron.

**3) What is a Concave Polyhedron?**

A polyhedron whose faces, edges and vertices intersect is called a concave polyhedron.

**Regular Polyhedron**

A regular polyhedron is made of regular polygons that meet on a vertex keeping the number of faces the same.

Shapes Name | Shape Figure | Faces |
---|---|---|

1. Cube | 6 square faces | |

2. Tetrahedron | 4 equilateral triangular faces | |

3. Octahedron | 8 equilateral triangular faces | |

4. Dodecahedron | 12 pentagonal faces | |

5. Icosahedron | 20 equilateral triangular faces |

**4) How Many Faces Does a Dodecahedron Have?**

A regular dodecahedron is made of 12 pentagonal faces.

**5) Name the Regular Polyhedron With a Square Face.**

A cube has 4 square faces.

**Euler’s Formula:**

According to Euler’s Formula, for any convex polyhedron, we have,

F + V – E = 2, where

F = Number of faces.

V = Number of vertices.

E = Number of edges.

Also, from Euler’s Formula, we get,

E = F + V – 2

F = 2 + E – V

V = 2 + E – F

**Number Operations**

**6) What Is the Difference Between the Regular Octahedron and the Tetrahedron?**

A regular octahedron comprises eight equilateral triangular faces, and a regular tetrahedron has four equilateral triangular faces.

**7) The Dodecahedron Has 20 Vertex Corners and 30 Edges. Find the Number of Faces It Has.**

From Euler’s formula, we have F = 2 + E – V

Substitute 20 for V and 30 for E in the formula F = 2 + E – V and then simplify to get the number of faces of the Dodecahedron.

F = 2 + 30 – 20

= 32 – 20

= 12

Therefore, Dodecahedron has 12 faces.

**8) Name the Regular Polyhedron, Which Has 20 Triangular Faces.**

A regular polyhedron that has 20 equilateral triangular faces is called an icosahedron.

**9) The Icosahedron Has 20 Triangular Faces and 30 Edges. Find the Number of Vertices It Has.**

From Euler’s formula, we have V = 2 + E – F

Substitute 20 for F and 30 for E in the formula V = 2 + E – F and then simplify to get the number of vertices of the Icosahedron.

V = 2 + 30 – 20

= 32 – 20

= 12

Therefore, Icosahedron has 12 vertices.

**10) The Dodecahedron Has 20 Vertex Corners and 30 Edges. Find the Number of Faces It Has.**

From Euler’s formula, we have F = 2 + E – V

Substitute 20 for V and 30 for E in the formula F = 2 + E – V and then simplify to get the number of faces of the Dodecahedron.

F = 2 + 30 – 20

= 32 – 20

= 12

Therefore, Dodecahedron has 12 faces.

**11) The Octahedron Has 6 Vertex Corners and 8 Faces. Find the Number of Edges It Has.**

From Euler’s formula, we have E = F + V – 2.

Substitute 8 for F and 6 for V in the formula E = F + V – 2 and then simplify to get the number of edges of the Octahedron.

E = 8 + 6 – 2

= 14 – 2

= 12

Therefore, Octahedron has 12 edges.

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